Indifferential geometry, thecotangent space is avector space associated with a point${\displaystyle x}$ on asmooth (or differentiable) manifold${\displaystyle \mathcal{M}}$; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space,${\displaystyle T_x^{\ast }\mathcal{M}}$ is defined as thedual space of thetangent space at${\displaystyle x}$,${\displaystyle T_x\mathcal{M}}$, although there are more direct definitions (seebelow). The elements of the cotangent space are calledcotangent vectors ortangent covectors.

All cotangent spaces at points on a connected manifold have the samedimension, equal to the dimension of the manifold. All the cotangent spaces of a manifold can be "glued together" (i.e. unioned and endowed with a topology) to form a new differentiable manifold of twice the dimension, thecotangent bundle of the manifold.

The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and thereforeisomorphic to each other via many possible isomorphisms. The introduction of aRiemannian metric or asymplectic form gives rise to anatural isomorphism between the tangent space and the cotangent space at a point, associating to any tangent covector a canonical tangent vector.

Let${\displaystyle \mathcal{M}}$ be a smooth manifold and let${\displaystyle x}$ be a point in${\displaystyle \mathcal{M}}$. Let${\displaystyle T_x\mathcal{M}}$ be thetangent space at${\displaystyle x}$. Then the cotangent space at${\displaystyle x}$ is defined as thedual space of${\displaystyle T_x\mathcal{M}}$:

${\displaystyle T_x^{\ast }\mathcal{M}=(T_x\mathcal{M}{)}^{\ast }}$

Concretely, elements of the cotangent space arelinear functionals on${\displaystyle T_x\mathcal{M}}$. That is, every element${\displaystyle \alpha \in T_x^{\ast }\mathcal{M}}$ is alinear map

${\displaystyle \alpha :T_x\mathcal{M}\to F}$

where${\displaystyle F}$ is the underlyingfield of the vector space being considered, for example, the field ofreal numbers. The elements of${\displaystyle T_x^{\ast }\mathcal{M}}$ are called cotangent vectors.

In some cases, one might like to have a direct definition of the cotangent space without reference to the tangent space. Such a definition can be formulated in terms ofequivalence classes of smooth functions on${\displaystyle \mathcal{M}}$. Informally, we will say that two smooth functionsf andg are equivalent at a point${\displaystyle x}$ if they have the same first-order behavior near${\displaystyle x}$, analogous to their linear Taylor polynomials; two functionsf andg have the same first order behavior near${\displaystyle x}$ if and only if the derivative of the functionf −g vanishes at${\displaystyle x}$. The cotangent space will then consist of all the possible first-order behaviors of a function near${\displaystyle x}$.

Let${\displaystyle \mathcal{M}}$ be a smooth manifold and let${\displaystyle x}$ be a point in${\displaystyle \mathcal{M}}$. Let${\displaystyle I_x}$be theideal of all functions in${\displaystyle C^{\mathrm{\infty }}(\mathcal{M})}$ vanishing at${\displaystyle x}$, and let${\displaystyle I_x^2}$ be the set of functions of the form$\sum _i{f}_i{g}_i$, where${\displaystyle f_i,g_i\in I_x}$. Then${\displaystyle I_x}$ and${\displaystyle I_x^2}$ are both real vector spaces and the cotangent space can be defined as thequotient space${\displaystyle T_x^{\ast }\mathcal{M}=I_x/I_x^2}$ by showing that the two spaces areisomorphic to each other.

This formulation is analogous to the construction of the cotangent space to define theZariski tangent space in algebraic geometry. The construction also generalizes tolocally ringed spaces.

Let${\displaystyle M}$ be a smooth manifold and let${\displaystyle f\in C^{\mathrm{\infty }}(M)}$ be asmooth function. The differential of${\displaystyle f}$ at a point${\displaystyle x}$ is the map

${\displaystyle \mathrm{d}{f}_x(X_x)=X_x(f)}$

where${\displaystyle X_x}$ is atangent vector at${\displaystyle x}$, thought of as a derivation. That is${\displaystyle X(f)={\mathcal{L}}_{X}f}$ is theLie derivative of${\displaystyle f}$ in the direction${\displaystyle X}$, and one has${\displaystyle \mathrm{d}f(X)=X(f)}$. Equivalently, we can think of tangent vectors as tangents to curves, and write

${\displaystyle \mathrm{d}{f}_x({\gamma }^{\prime }(0))=(f\circ \gamma {)}^{\prime }(0)}$

In either case,${\displaystyle \mathrm{d}{f}_x}$ is a linear map on${\displaystyle T_{x}M}$ and hence it is a tangent covector at${\displaystyle x}$.

We can then define the differential map${\displaystyle \mathrm{d}:C^{\mathrm{\infty }}(M)\to T_x^{\ast }(M)}$ at a point${\displaystyle x}$ as the map which sends${\displaystyle f}$ to${\displaystyle \mathrm{d}{f}_x}$. Properties of the differential map include:

1. ${\displaystyle \mathrm{d}}$ is a linear map:${\displaystyle \mathrm{d}(af+bg)=a\mathrm{d}f+b\mathrm{d}g}$ for constants${\displaystyle a}$ and${\displaystyle b}$,
2. ${\displaystyle \mathrm{d}(fg{)}_x=f(x)\mathrm{d}{g}_x+g(x)\mathrm{d}{f}_x}$

The differential map provides the link between the two alternate definitions of the cotangent space given above. Since for all${\displaystyle f\in I_x^2}$ there exist${\displaystyle g_i,h_i\in I_x}$ such that$f=\sum _i{g}_i{h}_i$, we have,

So that all function in${\displaystyle I_x^2}$ have differential zero, it follows that for every two functions${\displaystyle f\in I_x^2}$,${\displaystyle g\in I_x}$, we have${\displaystyle \mathrm{d}(f+g)=\mathrm{d}(g)}$. We can now construct anisomorphism between${\displaystyle T_x^{\ast }\mathcal{M}}$ and${\displaystyle I_x/I_x^2}$ by sending linear maps${\displaystyle \alpha }$ to the corresponding cosets${\displaystyle \alpha +I_x^2}$. Since there is a unique linear map for a given kernel and slope, this is an isomorphism, establishing the equivalence of the two definitions.

Just as every differentiable map${\displaystyle f:M\to N}$ between manifolds induces a linear map (called thepushforward orderivative) between the tangent spaces

${\displaystyle f_{\ast }^{}:T_{x}M\to T_{f(x)}N}$

every such map induces a linear map (called thepullback) between the cotangent spaces, only this time in the reverse direction:

${\displaystyle f^{\ast }:T_{f(x)}^{\ast }N\to T_x^{\ast }M.}$

The pullback is naturally defined as the dual (or transpose) of thepushforward. Unraveling the definition, this means the following:

${\displaystyle (f^{\ast }\theta )(X_x)=\theta (f_{\ast }^{}{X}_x),}$

where${\displaystyle \theta \in T_{f(x)}^{\ast }N}$ and${\displaystyle X_x\in T_{x}M}$. Note carefully where everything lives.

If we define tangent covectors in terms of equivalence classes of smooth maps vanishing at a point then the definition of the pullback is even more straightforward. Let${\displaystyle g}$ be a smooth function on${\displaystyle N}$ vanishing at${\displaystyle f(x)}$. Then the pullback of the covector determined by${\displaystyle g}$ (denoted${\displaystyle \mathrm{d}g}$) is given by

${\displaystyle f^{\ast }\mathrm{d}g=\mathrm{d}(g\circ f).}$

That is, it is the equivalence class of functions on${\displaystyle M}$ vanishing at${\displaystyle x}$ determined by${\displaystyle g\circ f}$.

The${\displaystyle k}$-thexterior power of the cotangent space, denoted${\displaystyle {\mathrm{\Lambda }}^k(T_x^{\ast }\mathcal{M})}$, is another important object in differential and algebraic geometry. Vectors in the${\displaystyle k}$-th exterior power, or more precisely sections of the${\displaystyle k}$-th exterior power of thecotangent bundle, are calleddifferential${\displaystyle k}$-forms. They can be thought of as alternating,multilinear maps on${\displaystyle k}$ tangent vectors. For this reason, tangent covectors are frequently calledone-forms.

• Abraham, Ralph H.;Marsden, Jerrold E. (1978),Foundations of mechanics, London: Benjamin-Cummings,ISBN 978-0-8053-0102-1
• Jost, Jürgen (2005),Riemannian Geometry and Geometric Analysis (4th ed.), Berlin, New York:Springer-Verlag,ISBN 978-3-540-25907-7
• Lee, John M. (2003),Introduction to smooth manifolds, Springer Graduate Texts in Mathematics, vol. 218, Berlin, New York:Springer-Verlag,ISBN 978-0-387-95448-6
• Misner, Charles W.;Thorne, Kip;Wheeler, John Archibald (1973),Gravitation, W. H. Freeman,ISBN 978-0-7167-0344-0

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